We study relationships between different relaxed notions of core stability in hedonic games, which are a class of coalition formation games. Our unified approach applies to a newly introduced family of hedonic games, called $\alpha$-hedonic games, which contains previously studied variants such as fractional and additively separable hedonic games. In particular, we derive an upper bound on the maximum factor with which a blocking coalition of a certain size can improve upon an outcome in which no deviating coalition of size at most $q$ exists. Counterintuitively, we show that larger blocking coalitions might sometimes have lower improvement factors. We discuss the tightness conditions of our bound, as well as its implications on the price of anarchy of core relaxations. Our general result has direct implications for several well-studied classes of hedonic games, allowing us to prove two open conjectures by Fanelli et al. (2021) for fractional hedonic games.

翻译：本研究探讨享乐博弈（一类联盟形成博弈）中不同松弛化核心稳定性概念之间的关系。我们提出的统一方法适用于新引入的$\alpha$-享乐博弈族，该族包含先前研究的多种变体，如分数享乐博弈与可加可分享乐博弈。特别地，我们推导了特定规模阻挡联盟在无规模不超过$q$的偏离联盟存在的博弈结果中可能获得的最大改进因子上界。反直觉的是，我们发现规模更大的阻挡联盟有时可能具有更低的改进因子。我们讨论了所得上界的紧性条件及其对核心松弛条件下无政府状态价格的影响。该通用结论对多个已被深入研究的享乐博弈类具有直接意义，使我们能够证明Fanelli等人（2021）针对分数享乐博弈提出的两个公开猜想。